The Turkish striker has been heavily linked with a move abroad this summer, after a scintillating debut campaign at the Turk Telekom Arena.
Gunners boss Arsene Wenger has been credited with an interest in bringing Yilmaz to the Emirates Stadium, as he attempts to fill the void left by the departed Robin van Persie.
Lukas Podolski and Olivier Giroud were brought in by Arsenal last summer, but neither have hit the sort of goalscoring form that had been anticipated.
Yilmaz scored 24 goals in 30 appearances in the Turkish top-flight in 2012/13, adding to the staggering 55 goals in 74 matches he scored for former club Trabzonspor in the two years previous to his switch to Galatasaray.
Speaking to medyafaresi.com, the 27-year-old said: "I am happy here in Turkey, I am comfortable and proud.
"The market and the transfers are also based on luck. I really like the football in England and Italy."
A move to north London would likely appeal to Yilmaz, with the chance to test himself in the Premier League an opportunity that could prove too good to turn down.